Natural Base (e) and Natural Logarithm (ln)

Hi guys! Here’s a short review on Natural Logarithms.

• Natural Base

The natural base ‘e’ is another strange number frequently used by  mathematicians which approximately equals to 2.71828

2.718281828x = ex

• Natural Logarithm

Natural Logarithms comes from the fancy Latin name Logarithmus naturali , giving the abbreviation ln. Natural Logarithms is really just the inverse of ‘e’.

e^x = y  —>  lne y = x

or

e^y = x   —>  ln x = y

Understanding Natural Logarithms

Reminder:  All Natural Logarithms use the same patterns and laws as Common Logarithms.

Expanding Natural Logarithmic Expressions

Roots—> Fraction Exponents Division—> Subtraction

Multiplication —> Addition Exponents —> Move to the front

Expand.

Follow expansion rules.

Do not forget to apply negative through.

Simplifying Natural Logarithmic Expressions

Rearrange terms (all negative terms  must be on the end) Fraction Exponents—> Roots

Addition —> Multiplication Move to the front —> Exponents Subtraction —> Division

Simplify.

Rearrange.

Do not forget to FACTOR OUT the negative.

Follow the simplification rules.

Solving Natural Logarithmic Equations

Simplify Convert Logarithm into exponential form (use 7 rule) Solve for x or any missing variable Check answers by plugging answer(s) back into the equation.

Place one logarithm with base e on each side of the eq’n. Using the Laws of Logarithms turn the exponent into a coefficient to the front of its logarithm. If the exponent contains a binomial, use brackets around the exponent. If the exponent is in brackets, multiply through brackets. Collect logarithms on one side and isolate the variable on the other. If there is more than one term containing the missing variable isolate the variable. Solve for the variable with your calculator Check your answer.

Evaluate. Round answer to three decimal place

Follow simplification rules. Rearrange.

Simplify further.

Convert to exponential form using 7 rule.

Solve for x.

Solving Using Cancellation Property

Evaluate

Take the left and right hand side of the equation and raise it as a power of e.

Cancel out same bases.

Raise left and right hand side as a power of e one last time.

Cancel out same bases.